Our objective is to maximise output within a cost constraint. Clearly, there are two parts of this problem – first, maximisation of output and secondly, satisfying the cost constraint. In order to fulfill the cost constraint, we can at most select a point on the isocost line, but cannot select a point right to it.
Secondly, maximisation of output is established if the highest possible indifference curve can be attained. For satisfying both the points simultaneously, we need to search for the highest isoquant which will be tangential to the isocost line.
Tangency ensures attainment of highest isoquant subject to a cost constraint. In other words, the isoquants, which are not tangent to the isocost line, are unable to fulfill both the conditions simultaneously.
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To explain this point, let us consider three isoquants I, II and III, out of which isoquant II is tangent to the iso-cost AB. Isoquant III is located right to it and isoquant I is located left of isoquant II.
Isoquant III (or any isoquant higher than isoquant II) yields more output than isoquant II and hence is preferred to isoquant II, but it violates cost constraint since it lies at the right side of the iso-cost line. So isoquant III is unattainable.
Isoquant I (or any isoquant lower than isoquant II) satisfies the cost constraint but yields less production than isoquant II. Thus, isoquant II which is tangent to isocost is the only one that satisfies both the conditions.
Mathematically, maximisation of production within a cost constraint requires one condition to be fulfilled – the slope of isoquant should be equal to the slope of isocost curve.
In figure 8.17, movement from point A to B indicates ‘loss’ of output arising from withdrawal of capital by ∆K which is just compensated by ‘gain’ in output by increasing labour input by ∆L, since A and B yield same quantity of output. Thus, loss in total productive capacity of ∆K amount of capital, i.e., ∆K × MPK must be equal to gain in total productive capacity of ∆L quantity of labour i.e., ∆L x MPL.
Again, MRTS measures the slope of AB segment of the given isoquant. If distance between A and B reduces, ultimately it will converge to a point. So, MRTS also measures slope of isoquent at a particular point.
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We have already mentioned that the point at which production of a given quantity of output is possible at least cost, must ensure equality between slope of isocost line (i.e., – w/r) and the slope of isoquant (i.e, MRTS) at that point.
The above condition ensures maximisation of output within the cost constraint.
Worked out Numerical Problem:
The production function is given as Q = K2 + L2 where w = Rs. 4 and r = Rs. 6. If the total cost of the firm is Rs. 780, calculate the maximum number of units that can be produced within the cost constraint.
